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Simplifying -8k2 + 8k + 2 = 0 Reorder the terms: 2 + 8k + -8k2 = 0 Solving 2 + 8k + -8k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '2'. 2(1 + 4k + -4k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(1 + 4k + -4k2)' equal to zero and attempt to solve: Simplifying 1 + 4k + -4k2 = 0 Solving 1 + 4k + -4k2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.25 + -1k + k2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -1k + 0.25 + k2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -1k + k2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -1k + k2 = 0 + 0.25 -1k + k2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -1k + k2 = 0.25 The k term is -1k. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1k + 0.25 + k2 = 0.25 + 0.25 Reorder the terms: 0.25 + -1k + k2 = 0.25 + 0.25 Combine like terms: 0.25 + 0.25 = 0.5 0.25 + -1k + k2 = 0.5 Factor a perfect square on the left side: (k + -0.5)(k + -0.5) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (k + -0.5) equal to 0.707106781 and -0.707106781.Subproblem 1
k + -0.5 = 0.707106781 Simplifying k + -0.5 = 0.707106781 Reorder the terms: -0.5 + k = 0.707106781 Solving -0.5 + k = 0.707106781 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + k = 0.707106781 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + k = 0.707106781 + 0.5 k = 0.707106781 + 0.5 Combine like terms: 0.707106781 + 0.5 = 1.207106781 k = 1.207106781 Simplifying k = 1.207106781Subproblem 2
k + -0.5 = -0.707106781 Simplifying k + -0.5 = -0.707106781 Reorder the terms: -0.5 + k = -0.707106781 Solving -0.5 + k = -0.707106781 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + k = -0.707106781 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + k = -0.707106781 + 0.5 k = -0.707106781 + 0.5 Combine like terms: -0.707106781 + 0.5 = -0.207106781 k = -0.207106781 Simplifying k = -0.207106781Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.207106781, -0.207106781}Solution
k = {1.207106781, -0.207106781}
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